Frames of Reference and Relative Velocity

Transcription

1 1.5 frame of reference coordinate ytem relative to which motion i oberved Frame of Reference and Relative Velocity Air how provide element of both excitement and danger. When high-peed airplane fly in contant formation (Figure 1), oberver on the ground ee them moving at high velocity. Seen from the cockpit, however, all the plane appear to have zero velocity. Oberver on the ground are in one frame of reference, while the pilot are in the plane frame of reference. A frame of reference i a coordinate ytem relative to which motion i decribed or oberved. The mot common frame of reference that we ue a a tationary, or fixed, frame of reference i arth or the ground. In the example of motion preented in the previou ection, all object were aumed to be moving relative to the frame of reference of arth. Sometime, however, other frame are choen for convenience. For example, to analyze the motion of the planet of the olar ytem, the Sun frame of reference i ued. If we oberve a pot near the rim of a rolling wheel, the wheel or the centre of the wheel i the mot convenient frame of reference, a in Figure 2. (a) (b) Figure 1 The Canadian Force Snowbird fly at velocitie of between 400 and 600 km/h (relative to the ground), but when they are flying in formation, a hown here, the velocity of one plane relative to another i zero. Figure 2 (a) The motion of a pot near the rim of a rolling wheel i imple if viewed from the frame of reference of the wheel centre. (b) The motion of the pot i much more complex when viewed from arth frame of reference. DID YOU KOW? Viewing the Solar Sytem It i eay to viualize planet revolving around the Sun, uing the Sun a the frame of reference. Ancient atronomer, however, ued arth frame of reference to try to explain the oberved motion of the planet, but had to invent force that do not exit. For example, when watching the motion of a planet beyond arth (uch a Mar) againt the background of the tar, the planet appear to revere direction from time to time, much like a flattened S pattern. In fact, the planet doen t revere direction; it only appear to do o a arth, which i cloer to the Sun, catche up and then pae the planet. The velocity of an object relative to a pecific frame of reference i called relative velocity. We have not ued thi term previouly becaue we were conidering motion relative to one frame of reference at a time. ow we will explore ituation involving at leat two frame of reference. Such ituation occur for paenger walking about in a moving train, for watercraft travelling on a flowing river, and for the Snowbird or other aircraft flying when there i wind blowing relative to the ground. To analyze relative velocity in more than one frame of reference, we ue the ymbol for relative velocity,, with two ubcript in capital letter. The firt ubcript repreent the object whoe velocity i tated relative to the object repreented by the econd ubcript. In other word, the econd ubcript i the frame of reference. For example, if P i a plane travelling at 490 km/h [W] relative to arth frame of reference,, then P 490 km/h [W]. If we conider another frame of reference, uch a the wind or air, A, affecting the plane motion, then PA i the velocity of the plane relative to the air and A i the velocity of the air relative to arth. The vector PA and A are related to P uing the following relative velocity equation: P PA A relative velocity velocity of an object relative to a pecific frame of reference Thi equation applie whether the motion i in one, two, or three dimenion. For example, conider the one-dimenional ituation in which the wind and the plane are both moving eatward. If the plane velocity relative to the air i 430 km/h [], and the air 52 Chapter 1 L

2 Section 1.5 velocity relative to the ground i 90 km/h [], then the velocity of the plane relative to the ground i: P PA A 430 km/h [] 90 km/h [] P 520 km/h [] Thu, with a tail wind, the ground peed increae a logical reult. You can eaily figure out that the plane ground peed in thi example would be only 340 km/h [] if the wind were a head wind (i.e., if AG = 90 km/h [W]). Before looking at relative velocitie in two dimenion, make ure that you undertand the pattern of the ubcript ued in any relative velocity equation. A hown in Figure 3, the left ide of the equation ha a ingle relative velocity, while the right ide ha the vector addition of two or more relative velocitie. ote that the outide and the inide ubcript on the right ide are in the ame order a the ubcript on the left ide. P = PA + A C = CW + W LO = LM + M + O Figure 3 The pattern in relative velocity equation SAMPL problem 1 DG = D + F + FG DID YOU KOW? Wind Direction By convention, a wet wind i a wind that blow from the wet, o it velocity vector point eat (e.g., a wet wind might be blowing at 45 km/h []). A outhwet wind ha the direction [45 of ] or [45 of ]. DID YOU KOW? avigation Terminology Air navigator have term for ome of the key concept of relative velocity. Air peed i the peed of a plane relative to the air. Wind peed i the peed of the wind relative to the ground. Ground peed i the peed of the plane relative to the ground. The heading i the direction in which the plane i aimed. The coure, or track, i the path relative to arth or the ground. Marine navigator ue heading, coure, and track in analogou way. An Olympic canoeit, capable of travelling at a peed of 4.5 m/ in till water, i croing a river that i flowing with a velocity of 3.2 m/ []. The river i m wide. (a) If the canoe i aimed northward, a in Figure 4, what i it velocity relative to the hore? (b) How long doe the croing take? (c) Where i the landing poition of the canoe relative to it tarting poition? (d) If the canoe landed directly acro from the tarting poition, at what angle would the canoe have been aimed? north hore w cw c river Figure 4 The ituation Solution Uing the ubcript C for the canoe, S for the hore, and W for the water, the known relative velocitie are: CW 4.5 m/ [] WS 3.2 m/ [] L Kinematic 53

3 LARIG TIP Alternative Symbol An alternative method of writing a relative velocity equation i to place the ubcript for the oberved object before the and the ubcript for the frame of reference after the. Uing thi method, the equation for our example of plane and air i P P A A. (a) Since the unknown i CS, we ue the relative velocity equation CS CW WS CS 4.5 m/ [] 3.2 m/ [] Applying the law of Pythagora, we find: CS (4.5 /) m 2 /) (3.2 m 2 CS 5.5 m/ Trigonometry give the angle v in Figure 4: v tan m/ 4. 5 m/ v 35 The velocity of the canoe relative to the hore i 5.5 m/ [35 of ]. (b) To determine the time taken to cro the river, we conider only the motion perpendicular to the river. d m [] CW 4.5 m/ [] t? From CW d, we have: t d t CW m [ ] 4. 5m/ [] t 49 The croing time i (c) The current carrie the canoe eatward (downtream) during the time it take to cro the river. The downtream diplacement i d WS t (3.2 m/ [])(49 ) d m [] The landing poition i m [] and m [] of the tarting poition. Uing the law of Pythagora and trigonometry, the reultant diplacement i m [36 of ]. (d) The velocity of the canoe relative to the water, CW, which ha a magnitude of 4.5 m/, i the hypotenue of the triangle in Figure 5. The reultant velocity CS mut point directly north for the canoe to land directly north of the tarting poition. north hore w cw v c f Figure 5 The olution for part (d) 54 Chapter 1 L

4 Section 1.5 The angle in the triangle i f in 1 WS CW in m/ 4. 5 m/ f 45 The required heading for the canoe i [45 W of ]. SAMPL problem 2 The air peed of a mall plane i 215 km/h. The wind i blowing at 57 km/h from the wet. Determine the velocity of the plane relative to the ground if the pilot keep the plane aimed in the direction [34 of ]. Solution We ue the ubcript Pfor the plane, for arth or the ground, and A for the air. PA 215 km/h [34 of ] A 57 km/h [] P? P PA A Thi vector addition i hown in Figure 6. We will olve thi problem by applying the coine and ine law; however, we could alo apply a vector cale diagram or component a decribed in Appendix A. Uing the coine law: P 2 PA 2 A 2 2 PA A co f (215 km/h) 2 (57 km/h) 2 2(215 km/h)(57 km/h) co 124 P 251 km/h 34 PA P f A f = f = 124 Figure 6 Solving Sample Problem 2 uing trigonometry Uing the ine law: in v in f A P 57 km/h (in 124 ) in v 251 km/h v 11 The direction of P i of. Thu P 251 km/h [45 of ]. Sometime it i helpful to know that the velocity of object X relative to object Y ha the ame magnitude a the velocity of Y relative to X, but i oppoite in direction: XY YX. Conider, for example, a jogger J running pat a peron P itting on a park bench. If JP 2.5 m/ [], then P i viewing J moving eatward at 2.5 m/. To J, P appear to be moving at a velocity of 2.5 m/ [W]. Thu PJ 2.5 m/ [] 2.5 m/ [W]. In the next Sample Problem, we will ue thi relationhip for performing a vector ubtraction. LARIG TIP Subtracting Vector When a relative velocity equation, uch a P PA A, i rearranged to iolate either PA or A, a vector ubtraction mut be performed. For example, PA P A i equivalent to PA P ( A ). Appendix A dicue vector arithmetic. L Kinematic 55

5 cale: 1.0 cm = 30 km/h SAMPL problem 3 A helicopter, flying where the average wind velocity i 38 km/h [25 of ], need to achieve a velocity of 91 km/h [17 W of ] relative to the ground to arrive at the detination on time, a hown in Figure 7. What i the neceary velocity relative to the air? HG = 91 km/h [17 W of ] Solution Uing the ubcript H for the helicopter, G for the ground, and A for the air, we have the following relative velocitie: AG = 38 km/h [25 of ] Figure 7 Situation for Sample Problem 3 HG 91 km/h [17 W of ] AG 38 km/h [25 of ] HA? HG HA AG We rearrange the equation to olve for the unknown: HA HG AG HA HG ( AG ) where AG i 38 km/h [25 S of W] cale: 1.0 cm = 30 km/h v AG HA HG Figure 8 how thi vector ubtraction. By direct meaurement on the cale diagram, we can ee that the velocity of the helicopter relative to the air mut be 94 km/h [41 W of ]. The ame reult can be obtained uing component or the law of ine and coine. Practice Undertanding Concept 1. Something i incorrect in each of the following equation. Rewrite each equation to how the correction. (a) L LD L Figure 8 Solution to Sample Problem 3 Anwer 2. (a) 3.9 m/ [fwd] (b) 1.7 m/ [fwd] (c) 3.0 m/ [21 right of fwd] m/ [12 of ] km [30 S of W] from Winnipeg (b) AC AB BC (c) M T TM (Write down two correct equation.) (d) LP ML M O OP 2. A cruie hip i moving with a velocity of 2.8 m/ [fwd] relative to the water. A group of tourit walk on the deck with a velocity of 1.1 m/ relative to the deck. Determine their velocity relative to the water if they are walking toward (a) the bow, (b) the tern, and (c) the tarboard. (The bow i the front of a hip, the tern i the rear, and the tarboard i on the right ide of the hip a you face the bow.) 3. The cruie hip in quetion 2 i travelling with a velocity of 2.8 m/ [] off the coat of Britih Columbia, in a place where the ocean current ha a velocity relative to the coat of 2.4 m/ []. Determine the velocity of the group of tourit in 2(c) relative to the coat. 4. A plane, travelling with a velocity relative to the air of 320 km/h [28 S of W], pae over Winnipeg. The wind velocity i 72 km/h [S]. Determine the diplacement of the plane from Winnipeg 2.0 h later. Making Connection 5. Airline pilot are often able to ue the jet tream to minimize flight time. Find out more about the importance of the jet tream in aviation. GO 56 Chapter 1 L

6 Section 1.5 SUMMARY Frame of Reference and Relative Velocity A frame of reference i a coordinate ytem relative to which motion can be oberved. Relative velocity i the velocity of an object relative to a pecific frame of reference. (A typical relative velocity equation i P PA A, where P i the oberved object and i the oberver or frame of reference.) Section 1.5 Quetion Undertanding Concept 1. Two kayaker can move at the ame peed in calm water. One begin kayaking traight acro a river, while the other kayak at an angle uptream in the ame river to land traight acro from the tarting poition. Aume the peed of the kayaker i greater than the peed of the river current. Which kayaker reache the far ide firt? xplain why. 2. A helicopter travel with an air peed of 55 m/. The helicopter head in the direction [35 of W]. What i it velocity relative to the ground if the wind velocity i (a) 21 m/ [] and (b) 21 m/ [22 W of ]? 3. A wimmer who achieve a peed of 0.75 m/ in till water wim directly acro a river 72 m wide. The wimmer land on the far hore at a poition 54 m downtream from the tarting point. (a) Determine the peed of the river current. (b) Determine the wimmer velocity relative to the hore. (c) Determine the direction the wimmer would have to aim to land directly acro from the tarting poition. 4. A pilot i required to fly directly from London, UK, to Rome, Italy in 3.5 h. The diplacement i km [43 of S]. A wind i blowing with a velocity of 75 km/h []. Determine the required velocity of the plane relative to the air. Applying Inquiry Skill 5. A phyic tudent on a train etimate the peed of falling raindrop on the train car window. Figure 9 how the tudent method of etimating the angle with which the drop are moving along the window gla. (a) Auming that the raindrop are falling traight downward relative to arth frame of reference, and that the peed of the train i 64 km/h, determine the vertical peed of the drop. (b) Decribe ource of error in carrying out thi type of etimation. direction of train motion left hand right hand Figure 9 timating the peed of falling raindrop Making Connection 6. You have made a video recording of a weather report, howing a reporter tanding in the wind and rain of a hurricane. How could you analyze the video to etimate the wind peed? Aume that the wind i blowing horizontally, and that the vertical component of the velocity of the raindrop i the ame a the vertical component for the raindrop in the previou quetion. L Kinematic 57